Filtering and Forecasting With Misspecified ARCH Models II

52 Pages·2017·1.19 MB·English

University of Pennsylvania
ScholarlyCommons
Statistics Papers Wharton Faculty Research
4-1992
Filtering and Forecasting With Misspecified ARCH
Models II: Making the Right Forecast With the
Wrong Model
Daniel B. Nelson
University of Pennsylvania
Dean P. Foster
University of Pennsylvania
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Recommended Citation
Nelson, D. B., & Foster, D. P. (1992). Filtering and Forecasting With Misspecified ARCH Models II: Making the Right Forecast With
the Wrong Model.Journal of Econometrics,52(1-2), 61-90.http://dx.doi.org/10.1016/0304-4076(92)90065-Y
This paper is posted at ScholarlyCommons.http://repository.upenn.edu/statistics_papers/584
For more information, please [email protected].
Filtering and Forecasting With Misspecified ARCH Models II: Making the
Right Forecast With the Wrong Model
Abstract
A companion paper (Nelson (1992)) showed that in data observed at high frequencies, an ARCH model may
do a good job at estimating conditional variances, even when the ARCH model is severely misspecified. While
such models may perform reasonably well atfiltering(i.e., at estimating unobserved instantaneous conditional
variances) they may perform disastrously at medium and long termforecasting. In this paper, we develop
conditions under which a misspecified ARCH model successfully performs both tasks, filtering and
forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH
model correctly specifies the functional form of the first two conditional moments of all state variables. We
apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility
model of Hull and White (1987), Scott (1987), and Wiggins (1987).
Disciplines
Economics | Statistics and Probability
This journal article is available at ScholarlyCommons:http://repository.upenn.edu/statistics_papers/584
NBER TECHNICAL WORKING PAPER SERIES
FILTERING AND FORECASTING
WITH MISSPECIFIED ARCH MODELS II:
MAKING THE RIGHT FORECAST
WITH THE WRONG MODEL
Daniel B. Nelson
Dean P. Foster
Technical Working Paper No. 132
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
December 1992
We thank Arnold Zellner and two anonymous referees for helpful comments. This material is
based on work supported by the National Science Foundation under grant #SES9110131. We
thank the Graduate School of Business of the University of Chicago, the Center for Research in
Security Prices, and the WilliamS. Fishman Research Scholarship for additional research support.
This paper is part of NBER's research program in Asset Pricing. Any opinions expressed are
those of the authors and not those of the National Bureau of Economic Research.
NBER Technical Working Paper #132
December 1992
FILTERING AND FORECASTING
WITH MISSPECIFIED ARCH MODELS II:
MAKING THE RIGHT FORECAST
WITH THE WRONG MODEL
ABSTRACT
A companion paper (Nelson (1992)) showed that in data observed at high frequencies, an ARCH
model may do a good job at estimating conditional variances, even when the ARCH model is
severely misspecified. While such models may perform reasonably well at filtering (i.e., at
estimating unobserved instantaneous conditional variances) they may perform disastrously at
medium and long term forecasting. In this paper, we develop conditions under which a
misspecified ARCH model successfully performs both tasks, filtering and forecasting. The key
requirement (in addition to the conditions for consistent filtering) is that the ARCH model
correctly specifies the functional form of the first two conditional moments of all state variables.
We apply these results to a diffusion model employed in the options pricing literature, the
stochastic volatility model of Hull and White (1987), Scott (1987), and Wiggins (1987).
Daniel B. Nelson Dean P. Foster
The University of Chicago The Wharton School
Graduate School of Business University of Pennsylvania
1101 East 58th Street Steinberg-Dietrich Hall
Chicago, IL 60637 3620 Locust Walk
and NBER Philadelphia, PA 19104
1. Introduction
Since their introduction by Engle (1982), ARCH models have been widely (and quite
successfully) applied in modeling financial time series; see, for example, the survey paper
of Bollerslev, Chou, and Kroner (1992). What accounts for the success of these models?
A companion paper (Nelson (1992)) suggested one reason: under fairly mild conditions, 1
high frequency data contain a great deal of information about conditional variances, and
as a continuous time limit is approached, the sample information about conditional
variances increases without bound. This allows simple volatility estimates formed, for
example, by taking a distributed lag of squared residuals (as in a GARCH model), to
consistently estimate conditional variances as the time interval between observations goes
to zero. It is not surprising, therefore, that as continuous time is approached, a sequence
of GARCH models can consistently estimate the underlying conditional variance of a
diffusion, even when the GARCH models are not the correct data generating process.
As shown in Nelson (1992), this continuous time consistency holds not only for
GARCI-1, but for many other ARCH models as well, and is unaffected by a wide variety
of misspecifications. For example, when considered as a data generating process, a given
ARCH model might provide nonsensical forecasts--for example, by forecasting a negative
conditional variance with positive probability, incorrectly forecasting explosions in the state
variables, ignoring unobservable state variables, or by misspecifying the conditional mean.
Nevertheless, in the limit as continuous time is approached, such a model can provide a
consistent estimate of the instantaneous conditional variance in the ~nderlying data
l) i.e .. if the observed data is generated by a diffusion or near-diffusion (i.e., by a stochastic
process imbedded in a sequence of processes converging weakly to a diffusion).
generating process. (Our use of the term "estimation" corresponds to its use in the filtering
literature rather in the statistics literature--i.e., the ARCH model "estimates" the conditional
variance in the sense that a Kalman filter estimates unobserved state variables. See, e.g.,
Anderson and Moore ( 1979, Chapter 2), or Arnold (1973, Chapter 12).) In other words,
while a misspecified ARCH model may perform disastrously in medium or long-term
forecasting, it may perform well at filtering.
In this paper. we show that under suitable conditions, a sequence of misspecified
ARCH models may not only be successful at filtering, but at forecasting as welL That is,
as a continuous time limit is approached, not only do the conditional covariance matrices
generated by the sequence of ARCH models approach the true conditional covariance
matrix. but the forecasts generated by these models converge in probability to the forecasts
generated by the true data generating process.
The conditions for consistent estimation of the conditional distribution are
considerably stricter than the conditions for consistent filtering; for example, all
unobservable state variables must be consistently estimated and we must correctly specify
the conditional mean and covariance of all state variables as continuous time is approached.
These conditions are developed in Section 2.
While the conditions in Section 2 are stricter than the consistent filtering conditions
developed in Nelson (1992), they are broad enough to accommodate a number of
interesting cases. In Section 3, we provide a detailed example, using a stochastic volatility
model familiar in the options pricing literature (see, e.g., Wiggins (1987), Hull and White
(1987), Scott (1987), and Melino and Turnbull (1990)). In this model, a stock price and its
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instantaneous returns volatility follow a diffusion process. While the stock price is
observable at discrete intervals of length h, the instantaneous volatility is unobservable. We
show that a suitably constructed sequence of ARCH models (in particular, AR(l)
EGARCH models) can consistently estimate the instantaneous volatility and generate
appropriate forecasts of the stock price and volatility processes as h~O.
2. Main results
In this section, we develop our basic results on consistent estimation of forecast
distributions. We begin by taking the nonnegative real line, chopping it up into pieces of
length h. and considering, for each h, a stochastic processes {h~•hU1} which is a step
function with jumps only (with probability one) at integer multiples of h--i.e., at times h,
2h. 3h. and so on. We interpret {hX1} as the nx 1 discretely observable component of the
process and C1 U1} as the m X 1 unobservable component. Associated with the true data
generating process is a probability measure Ph, which we define below.
The misspecified ARCH model produces an estimate h0 of the true unobservable
1
state variables 11U1. Associated with the ARCH model is a (misspecified) probability
measure i\ for {11~ q1 Ul'h 01}. The ARCH model incorporates the false assumption that
11 U1 = 1101 for all t almost surely. Our interest is in comparing forecasts made using
I\
information at a timeT with the incorrect probability measure to those made using the
correct measure P11• Specifically, how can we characterize the forecasts generated by Ph and
i\
as h~O? Under what circumstances do they become "close" as h~O? Theorems 2.3 and
2.4 below compare the conditional forecast distribwions generated by Ph and Ph, while
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Theorem 2.5 compares the conditional forecast moments. Finally, Theorem 2.6 compares
the very long term forecasts (i.e., the forecast stationary distributions) generated by Ph and
I\.
The Formal Setup
Let D([O,oo),R0xR2m) be the space of functions from [O,oo) into R0xR2m that are
continuous from the right with finite left limits. D is a metric space when endowed with the
Skorohod metric (see Ethier and Kurtz (1986, Chapter 3) for formal definitions). For each
h > 0, let hSfbe the a-algebra generated by h00, 11Xo, hXh, hXzh• ... hXkh• and hU0, hUh,
h U 211, ... hU kh for all kh :s t. In our notation, curly brackets indicate a stochastic process--e.g.,
{11~,hUt}[O.T] is the sample path of h~ and hUt as (random) functions of time on the
interval 0 :s t :s T. We refer to values of a process at a particular time t by omitting the
curly brackets--e.g., X, and ( X, , U ,) are, respectively, the (random) values taken by the
11 11 11
{11~} and {11~.11U,} processes at timer.
v
Next, let 23(E) denote the Borel sets on a metric space E, and let vh and h be
probability measures on (Rn+lm,23(R0XR2m)). Below, we will take vh and vh to be,
respectively, probability measures for the starting values (hXo•hU0,hU0) under the true data
generating process and under the misspecified ARCH model.
The functional limit theorems we employ will require { Xkh•h Ukh•h Ukh}k=O."' to have
11
i\.
a first order Markov structure under both Ph and Accordingly, our next step is to
introduce the Markov transition probabilities associated with. the true data generating
process and the misspecified ARCH model respectively: For each h > 0, let Ilh(x,u, ~ •. )and
4
IT (x,u,u,.) be transition functions on RnxR2m, i.e., IIh(x,u,u,·) and fih(x,u,u.-) are
11
probability measures on (RnxR2m,23(RnxR2m)) for all (x,u, u) E RnxR2m, and IIh(·.-.-,f)
and fth(-,·,·,f) are 23(RnxR2m) measurable for all r E 23(RnxR2m). Expectations
i\
evaluated under Ph and are respectively denoted Eh[ ·] and Eh[ ·].
In our examples, {hUkh} consists either of the unique elements of the estimated
conditional covariance matrix generated by an ARCH model or some invertible function
of these conditional covariances. In ARCH models, the conditional covariance matrix is a
function of past values of hXkh along with a startup value for the estimated conditional
covariance matrix. In particular, we assume that h Okh is a measurable function of h O(k·l)h,
11X(k-l)b, and hxkh. This in turn implies that given the observed data hXo• hxh, hX2h, ... hXkh
and the startup value 1100, we can derive the whole sequence {hUkh}j=l.k· Formally, we
require that for each h > 0, there is a measurable function Uh from R2n+m into Rrn such
that for all (x,u, u ),
f fth(x,u,u,d(x',u',u')) = f ITh(x,u,u,d(x',u',u')) 1. (1)
u) }
U11(x",x. { ~ • = U h (X • ,X, ~ ) }
v
We also assume that the misspecified ARCH model generated by and fi treats u and
11 11
~ as being identical almost surely--i.e., for all h > 0, and all (x,u, u) E Rn+2m
f ITJx,u,u,d(x',u',li')) 1, (2)
1
{u·=u·}
and
1. (3)
Finally, we assume that there is no feedback from the ARCH estimate {h Ukh }k=0.1.2. .. into
5
{hXkh·hukh}k=o.u ... --i.e., given hxkh and hukh, hX(k+J)h and 11U(k+l)h are independent of
hOkh under Ph. Formally, for any f E .23(Rn+m) and all h > 0, we require that
(4)
I\
where "001Xl" is an m x 1 vector of zeros. Now let P11 and be the probability measures
on D([O,oo),R0XR201) such that
Ph[ (hXo•h Uo•h Uo) E r l = vh(f) for any r E .23(R0 X R2m), (5)
l\[ (hXo·hUO•hUo) E f] = ~11(f) for any f E .23(R0xR2m), (5')
Ph[ (11X,.hU,.11U,) = (11Xkh•hUkh•h0kh), kh :5 t < (k+1)h] = 1, (6)
i\[ (h~·hut•hO,) = c,xkh•hukh•hOkhl· kh =o; t < (k+1)h l = 1, (6')
and finally, for all k ~ 0 and f E .23(R0XR2m),
Ph[ chx(k+l)ll'hu(k+l)h·ho<k+l)hl E r I hsrh J = rrhchxkh•hukh•hokh•r) (7)
almost surely under Ph, and
i\[ chx<k+l)h'i,u(k+l)h•ho(k+l)hl E r I hsrh l = i\chxkh•hukh•hokh•r) (7')
I\.
almost surely under
For each h > 0, (5) specifies the distributions of the starting point (hXo , hU 0, h0 0)
under Ph. (6) fonns the continuous time process {hXt•hUt•hOt}O:st as a step function with
jumps at times 0. h, 2h ..... 2 (7) specifies the transition probabilities for the jumps in
2) Our step function scheme follows Ethier and Kurtz (1986). Many other schemes would
work just as well--for example, we could follow Stroock and Varadhan (1979·, p. 267) and for
kh:5t<(k+l)h, set (hXt , hut , hOt) - (hXkh hukh , hOkh) +
h-1(t-kh)[(hX(k+t)h. hu(k+I)h, hO(k+l)h)-(hXkh , hukh , hOkh)J. This scheme makes
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